The temperature distribution θ(x, t) along an insulated metal rod of length L is described by the differential equation: ∂ (∂2θ/∂x2) = (1/D)(∂θ/∂t), (0 < x < L, t > 0) where D ̸= 0 is a constant. The rod is held at a fixed temperature of 0◦C atone end and is insulated at the other end, which gives rise to the boundary conditions (∂θ/∂x) = 0 when x = 0 for t > 0 together with θ = 0 when x = L for t > 0. The initial temperature distribution in the rod is given by: θ (x,0) = 0.3 cos (7/2)(πx/L), (0 ≤ x ≤ L) 1. Use the method of separation of variables, with θ (x,t) = X(x)T(t), to show that the function X(x) satisfies the differential equation X" - μX = 0 for some constant μ. Write down the corresponding differential equation that T(t) must satisfy. 2. Find the two boundary conditions that X(x) must satisfy.
The temperature distribution θ(x, t) along an insulated metal rod of length L is described by the differential equation: ∂
(∂2θ/∂x2) = (1/D)(∂θ/∂t), (0 < x < L, t > 0)
where D ̸= 0 is a constant. The rod is held at a fixed temperature of 0◦C atone end and is insulated at the other end, which gives rise to the boundary conditions (∂θ/∂x) = 0 when x = 0 for t > 0 together with θ = 0 when x = L for t > 0.
The initial temperature distribution in the rod is given by:
θ (x,0) = 0.3 cos (7/2)(πx/L), (0 ≤ x ≤ L)
1. Use the method of separation of variables, with θ (x,t) = X(x)T(t), to show that the function X(x) satisfies the differential equation
X" - μX = 0
for some constant μ. Write down the corresponding differential equation that T(t) must satisfy.
2. Find the two boundary conditions that X(x) must satisfy.
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